# Supporting Research, Evidence and Argument

Key texts for the key components of teaching for mastery

An exchange programme between teachers from England and Shanghai has informed effective pedagogic strategies for achieving mastery of maths. The striking performances of Shanghai and other East Asian countries in maths have become well-established. They have been measured in successive international tests such as TIMSS and PISA.

For each of the key components of teaching for mastery identified below, we offer a key text. This text is chosen for its relevance and accessibility, both in terms of readability and being able to access it for free online. We also provide a list of further relevant reading. Links in the main text will take you to the suggested key text and further reading. Much of this material features on a reading list for teachers on the Maths Hubs Mastery Specialists Programme.

The component thought to be key to the success of the East Asian system is the use of variation theory. Variation theory has several dimensions, including use of multiple representations of what a concept is, and what it is not. It is characterised by a carefully constructed small-step journey through learning. It pays attention to what is kept the same and what changes, in order that pupils might reason. This means that they will make connections and build deep conceptual knowledge.

Variation is applied to practice questions where attention is paid to the selection and order of the examples. Often just one aspect is changed whilst others are kept the same. The intention is to avoid mechanical repetition but instead to promote thinking to make connections. This is also known as ‘intelligent practice’.

Teaching for mastery is also characterised by a series of beliefs and practices.

### Children Succeeding

Mastery is characterised by a belief that, by working hard, all children are capable of succeeding at mathematics. On this basis, children are taught all together as a class and are not split into ‘prior attainment’ groupings.

Carefully structured teaching is planned in small steps. This provides both the necessary scaffold for all to achieve, and the necessary detail and rigour of all aspects of the maths to facilitate deep thinking. The small steps are connected and concepts are built. This leads to generalisation of the maths, and the ability to apply it to multiple contexts and solve problems.

It is expected that those children who will achieve well on a particular topic may not necessarily be the same children who achieved well on other topics. An additional daily short session of 10 to 15 minutes is provided for any pupils who do not fully grasp the lesson content, in order that they 'keep up' with the class. Our experience shows that it is not always the same pupils who require this form of intervention and this boosts the self-belief of previously low-attaining pupils.

### Understanding Structures

A focus on exposing the structure of mathematics and developing an understanding of how and why maths works is crucial to mastery. A key skill of the teacher is to be able to represent the maths in ways that provide access and insight for pupils.

Concrete materials, contexts, drawings, diagrams and equations all play a role. These are discussed through opportunities for pupil-pupil and pupil-teacher talk, to develop reasoning, flexibility and adaptability in mathematical thinking.

### Learning Facts

Memorisation and repetition of key facts (times tables and number bonds etc.) are important aspects of learning. Evidence from cognitive science research suggests that learning key facts so they can be recalled automatically ‘frees up’ working memory. It can then focus on more complex problem solving, rather than reaching cognitive overload trying to calculate simple operations. In terms of procedural fluency and conceptual understanding, one should not be prioritised over the other. Learning is most effective when the two are fully integrated.

### Mathematical Language

Teaching children precise mathematical language and insisting upon its use supports children's ability to think mathematically. Having the language and using it empowers children’s ability to think about the concept.

Teaching for mastery is in the early stages of adoption in England. We will be monitoring research and evidence of its effectiveness and adding to the final section, Recent evidence.

### Notes and further reading

This video summarises TIMSS international results for 2015. You can also read the report for England. You may also be interested in the report for PISA International results 2015, and the report for England.

**Key text **

Kullberg, Angelika, Ulla Runesson Kempe, and Ference Marton. "What is made possible to learn when using the variation theory of learning in teaching mathematics?." *ZDM* (2017): 1-11.

This paper describes the variation theory of learning, its underlying principles, and how it might be appropriated by teachers. It is illustrated by an analysis of one teacher’s teaching before and after he participated in three lesson studies based on variation theory.

**Further reading **

Sun, Xu Hua. “The structures, goals and pedagogies of ‘variation problems’” in the topic of addition and subtraction of 0-9 in Chinese textbooks and reference books." *Eighth Congress of European Research in Mathematics Education (CERME 8), Apr. 2013*. 2013.

This article explores Variation Theory drawing on examples from Chinese primary textbooks for the teaching of addition and subtraction of single digit numbers.

Watson, Anne, and John Mason. "Seeing an exercise as a single mathematical object: Using variation to structure sense-making." *Mathematical thinking and learning* 8.2 (2006): 91-111.

This article looks in depth at use of Variation Theory in written exercises given to learners and explains how exercises written using Variation Theory can support learners in perceiving relationships and structure rather than simply practising a mechanical procedure.

GU Lingyuan HUANG Rongjin MARTON, Ference. "Teaching with variation: A Chinese way of promoting effective mathematics learning." Ch.12 p309 of this book: Lianghuo, Fan, et al., eds. How Chinese learn mathematics: Perspectives from insiders. Vol. 1. World Scientific, 2004 (the online version is not complete and does not include Ch. 12).

Marton researches more and less successful instances of learning in various collaborations. In this paper, he collaborates with Chinese researchers to ‘unlock the paradox of the Chinese learner’ with variation theory as the key.

Askew, Mike. Transforming primary mathematics: understanding classroom tasks, tools and talk. Routledge, 2015. Ch. 6, p75 discusses Variation Theory (the online version is not complete and does not include Ch. 6).

Askew provides a readable account of how using Variation Theory can turn mathematics from a set of procedures to a ‘mindful activity’.

**Key text **

Dweck, Carol S. "Mindsets and math/science achievement." (2014).

Dweck is a cognitive scientist, known for her work on ‘Growth Mindset’ – the belief that ability is not fixed, but something that can be worked upon. Here, she applies her theories to maths and science in particular.

**Further reading **

Watson, Anne, Els De Geest, and Stephanie Prestage. "Deep Progress in Mathematics." *University of Oxford* (2003).

Project where a handful of teachers looked at ways to approach teaching bottom sets removing assumptions about ‘can’t’ (also available here).

Boaler, J. (1997) *Experiencing School Mathematics: Teaching Styles, Sex and Setting*. OUP.

This text has since been revised in 2002 for the American audience, but the original is still in print, though not available online.

Cambridge Mathematics Espresso 5 summarises recent research into ability-grouping in Maths.

The Education Endowment Fund considers the impact of setting/streaming in its Teaching and Learning Toolkit.

**Key text **

Ma, L. (1999). *Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States*. Mahwah, NJ: Lawrence Erlbaum Associates.

This research compares the ‘profound understanding of fundamental mathematics’ of teachers in China and the U.S., and how they would use this understanding to inform their teaching. The results demonstrate the deep structural understanding held by Chinese teachers that helps inform their teaching strategies. Summarised and reviewed here - book is in print, but not available online.

**Key text **

Willingham, Daniel T. "Is it true that some people just can’t do math?" *American Educator* 33.4 (2009): 14-19.

This paper offers evidence from cognitive science of cognitive overload. It argues that that learning facts so they can be recalled automatically frees up working memory, so avoids cognitive overload. Willingham also argues that procedural fluency and conceptual understanding should be taught in tandem.

**Further reading **

Cambridge Mathematics Espresso 10 asks Why is working memory important for mathematics learning?

Baroody, Arthur J. "Mastering the basic number combinations." *Teaching Children Mathematics 23* (2006): 22-31.

This article argues that basic number fluency is best achieved by teaching it alongside conceptual understanding. Includes classroom suggestions and examples.

Dahlin, Bo, and David Watkins. "The role of repetition in the processes of memorising and understanding: A comparison of the views of German and Chinese secondary school students in Hong Kong." British Journal of Educational Psychology 70.1 (2000): 65-84 (full text requires payment).

This research gathers and compares the experience of learning through repetition and memorisation, of German and Chinese students studying in Hong Kong. Results suggest that the Chinese students place more emphasis on the role that repetition plays in understanding than their German peers.

A guidance report from the Education Endowment Foundation Improving Mathematics in Key Stages 2 & 3 was published in November 2017. It endorses many of the components of teaching for mastery. Links between its recommendations and a teaching for mastery approach are made explicit in this blog by Professor Jeremy Hodgen. Professor Hodgen is Chair of Mathematics Education at the UCL Institute of Education, who led the evidence review.

A report by the Fair Education Alliance looks at schools with good outcomes for disadvantaged children in maths. Investigating 20 schools and Early Years providers, it pinpoints the factors in their success. Many of the schools cite teaching for mastery as a key factor.